BarotropicQGQL Module

Basic Equations

This module solves the quasi-linear quasi-geostrophic barotropic vorticity equation on a beta-plane of variable fluid depth $H-h(x,y)$. Quasi-linear refers to the dynamics that neglect the eddy–eddy interactions in the eddy evolution equation after an eddy–mean flow decomposition, e.g.,

\[\phi(x, y, t) = \overline{\phi}(y, t) + \phi'(x,y,t) ,\]

where overline above denotes a zonal mean, $\overline{\phi}(y, t) = \int \phi(x, y, t)\,\mathrm{d}x/L_x$, and prime denotes deviations from the zonal mean. This approximation is used in many process-model studies of zonation, e.g.,

Farrell, B. F. and Ioannou, P. J. (2003). Structural stability of turbulent jets. J. Atmos. Sci., 60, 2101-2118.
Srinivasan, K. and Young, W. R. (2012). Zonostrophic instability. J. Atmos. Sci., 69 (5), 1633-1656.
Constantinou, N. C., Farrell, B. F., and Ioannou, P. J. (2014). Emergence and equilibration of jets in beta-plane turbulence: applications of Stochastic Structural Stability Theory. J. Atmos. Sci., 71 (5), 1818-1842.

As in the BarotropicQG module, the flow is obtained through a streamfunction $\psi$ as $(u, \upsilon) = (-\partial_y\psi, \partial_x\psi)$. All flow fields can be obtained from the quasi-geostrophic potential vorticity (QGPV). Here the QGPV is

\[\underbrace{f_0 + \beta y}_{\text{planetary PV}} + \underbrace{(\partial_x \upsilon - \partial_y u)}_{\text{relative vorticity}} + \underbrace{\frac{f_0 h}{H}}_{\text{topographic PV}}.\]

The dynamical variable is the component of the vorticity of the flow normal to the plane of motion, $\zeta\equiv \partial_x \upsilon- \partial_y u = \nabla^2\psi$. Also, we denote the topographic PV with $\eta\equiv f_0 h/H$. After we apply the eddy-mean flow decomposition above, the QGPV dynamics are:

\[\partial_t \overline{\zeta} + \mathsf{J}(\overline{\psi}, \underbrace{\overline{\zeta} + \overline{\eta}}_{\equiv \overline{q}}) + \overline{\mathsf{J}(\psi', \underbrace{\zeta' + \eta'}_{\equiv q'})} = \underbrace{-\left[\mu + \nu(-1)^{n_\nu} \nabla^{2n_\nu} \right] \overline{\zeta} }_{\textrm{dissipation}} \ .\]

\[\partial_t \zeta' + \mathsf{J}(\psi', \overline{q}) + \mathsf{J}(\overline{\psi}, q') + \underbrace{\mathsf{J}(\psi', q') - \overline{\mathsf{J}(\psi', q')}}_{\textrm{EENL}} + \beta\partial_x\psi' = \underbrace{-\left[\mu + \nu(-1)^{n_\nu} \nabla^{2n_\nu} \right] \zeta'}_{\textrm{dissipation}} + f\ .\]

where $\mathsf{J}(a, b) = (\partial_x a)(\partial_y b)-(\partial_y a)(\partial_x b)$. On the right hand side, $f(x,y,t)$ is forcing (which is assumed to have zero mean, $\overline{f}=0$), $\mu$ is linear drag, and $\nu$ is hyperviscosity. Plain old viscosity corresponds to $n_{\nu}=1$. The sum of relative vorticity and topographic PV is denoted with $q\equiv\zeta+\eta$.

Quasi-linear dynamics neglect the term eddy-eddy nonlinearity (EENL) term above.

Implementation

The equation is time-stepped forward in Fourier space:

\[\partial_t \widehat{\zeta} = - \widehat{\mathsf{J}(\psi, q)}^{\textrm{QL}} +\beta\frac{\mathrm{i}k_x}{k^2}\widehat{\zeta} -\left(\mu +\nu k^{2n_\nu}\right) \widehat{\zeta} + \widehat{f}\ .\]

In doing so the Jacobian is computed in the conservative form: $\mathsf{J}(f,g) = \partial_y [ (\partial_x f) g] -\partial_x[ (\partial_y f) g]$. The superscript QL in the Jacobian term above denotes that remove triad interactions that correspond to the EENL term.

Thus:

\[\mathcal{L} = \beta\frac{\mathrm{i}k_x}{k^2} - \mu - \nu k^{2n_\nu}\ ,\]

\[\mathcal{N}(\widehat{\zeta}) = - \mathrm{i}k_x \mathrm{FFT}(u q)^{\textrm{QL}}- \mathrm{i}k_y \mathrm{FFT}(\upsilon q)^{\textrm{QL}}\ .\]

Examples

examples/barotropicqgql_betaforced.jl: A script that simulates forced-dissipative quasi-linear quasi-geostrophic flow on a beta-plane demonstrating zonation. The forcing is temporally delta-correlated and its spatial structure is isotropic with power in a narrow annulus of total radius kf in wavenumber space. This example demonstrates that the anisotropic inverse energy cascade is not required for zonation.